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''wreath of''
 
''wreath of''
  
Surfaces associated with an [[Infinitesimal deformation|infinitesimal deformation]] of one of them; discovered by G. Darboux [[#References|[1]]]. Darboux surfaces form a  "wreath"  of 12 surfaces, with radius vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301701.png" /> satisfying the equations
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Surfaces associated with an [[Infinitesimal deformation|infinitesimal deformation]] of one of them; discovered by G. Darboux [[#References|[1]]]. Darboux surfaces form a  "wreath"  of 12 surfaces, with radius vectors $  \mathbf x _ {1} \dots \mathbf x _ {6} , \mathbf z _ {1} \dots \mathbf z _ {6} $
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satisfying the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301702.png" /></td> </tr></table>
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$$
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d \mathbf z _ {i}  = [ \mathbf z _ {i + 1 }  , d \mathbf x _ {i} ] ,\ \
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d \mathbf x _ {i}  = [ \mathbf x _ {i - 1 }  , d \mathbf z _ {i} ] ,
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301703.png" /></td> </tr></table>
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$$
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\mathbf z _ {i} - \mathbf x _ {i + 1 }  = [ \mathbf z _ {i+ 1 }  , \mathbf x _ {i} ],\  i = 1 \dots 6 ,
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301704.png" /></td> </tr></table>
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$$
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\mathbf x _ {i + 6 }  = \mathbf x _ {i} ,\  \mathbf z _ {i + 6 }  = \mathbf z _ {i} ;
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301706.png" /> are in [[Peterson correspondence|Peterson correspondence]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301708.png" /> are in [[Polar correspondence|polar correspondence]], while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d0301709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d03017010.png" /> are poles of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d03017011.png" />-congruence. A similar  "wreath"  is formed by pairs of isometric surfaces of an elliptic space.
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where $  \mathbf z _ {i+} 1 $
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and $  \mathbf x _ {i} $
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are in [[Peterson correspondence|Peterson correspondence]], $  \mathbf z _ {i+} 1 $
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and $  \mathbf x _ {i-} 1 $
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are in [[Polar correspondence|polar correspondence]], while $  \mathbf z _ {i} $
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and $  \mathbf x _ {i+} 1 $
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are poles of a $  W $-
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congruence. A similar  "wreath"  is formed by pairs of isometric surfaces of an elliptic space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''4''' , Gauthier-Villars  (1896)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''4''' , Gauthier-Villars  (1896)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For the notion of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030170/d03017012.png" />-congruence cf. [[Congruence of lines|Congruence of lines]].
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For the notion of a $  W $-
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congruence cf. [[Congruence of lines|Congruence of lines]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Fubini,  E. Čech,  "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Fubini,  E. Čech,  "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars  (1931)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Bol,  "Projective Differentialgeometrie" , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR></table>

Latest revision as of 17:32, 5 June 2020


wreath of

Surfaces associated with an infinitesimal deformation of one of them; discovered by G. Darboux [1]. Darboux surfaces form a "wreath" of 12 surfaces, with radius vectors $ \mathbf x _ {1} \dots \mathbf x _ {6} , \mathbf z _ {1} \dots \mathbf z _ {6} $ satisfying the equations

$$ d \mathbf z _ {i} = [ \mathbf z _ {i + 1 } , d \mathbf x _ {i} ] ,\ \ d \mathbf x _ {i} = [ \mathbf x _ {i - 1 } , d \mathbf z _ {i} ] , $$

$$ \mathbf z _ {i} - \mathbf x _ {i + 1 } = [ \mathbf z _ {i+ 1 } , \mathbf x _ {i} ],\ i = 1 \dots 6 , $$

$$ \mathbf x _ {i + 6 } = \mathbf x _ {i} ,\ \mathbf z _ {i + 6 } = \mathbf z _ {i} ; $$

where $ \mathbf z _ {i+} 1 $ and $ \mathbf x _ {i} $ are in Peterson correspondence, $ \mathbf z _ {i+} 1 $ and $ \mathbf x _ {i-} 1 $ are in polar correspondence, while $ \mathbf z _ {i} $ and $ \mathbf x _ {i+} 1 $ are poles of a $ W $- congruence. A similar "wreath" is formed by pairs of isometric surfaces of an elliptic space.

References

[1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 4 , Gauthier-Villars (1896)
[2] V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)

Comments

For the notion of a $ W $- congruence cf. Congruence of lines.

References

[a1] G. Fubini, E. Čech, "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars (1931)
[a2] G. Bol, "Projective Differentialgeometrie" , Vandenhoeck & Ruprecht (1954)
How to Cite This Entry:
Darboux surfaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_surfaces&oldid=46581
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article